Problem: The lifespans of snakes in a particular zoo are normally distributed. The average snake lives $22.3$ years; the standard deviation is $4.8$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a snake living between $12.7$ and $27.1$ years.
$22.3$ $17.5$ $27.1$ $12.7$ $31.9$ $7.9$ $36.7$ $95\%$ $68\%$ $13.5\%$ $13.5\%$ We know the lifespans are normally distributed with an average lifespan of $22.3$ years. We know the standard deviation is $4.8$ years, so one standard deviation below the mean is $17.5$ years and one standard deviation above the mean is $27.1$ years. Two standard deviations below the mean is $12.7$ years and two standard deviations above the mean is $31.9$ years. Three standard deviations below the mean is $7.9$ years and three standard deviations above the mean is $36.7$ years. We are interested in the probability of a snake living between $12.7$ and $27.1$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the snakes will have lifespans within 2 standard deviations of the average lifespan. It also tells us that $68\%$ of the snakes will have lifespans within 1 standard deviation of the mean. The probability of a particular snake living between $12.7$ and $27.1$ years is $\color{orange}{13.5\%} + {68\%}$, or $81.5\%$.